Just the term "gifted child" is enough to raise eyebrows in disdain. Being gifted – having special talents above the average – has never been endearing. No matter how it's presented, it smacks of showing off, of thinking that you're better than anyone else. As with many other labels, it can be a burden.

"In this country," says Professor Joan Freeman, "the stereotype is usually of a little boy wearing glasses, can't make friends, verging on Asperger's, may play a violin, small, thin, friendless." Freeman used to subscribe to quite a lot of these myths. "But as the years went by, I began to realise that gifted children are normal human beings."

Freeman, a psychologist, knows more about this subject than most, having published a book about gifted children and how they grow up. In Gifted Lives, she chronicles the lives of 20 adults she has followed for 35 years since they were children, variously aged five to 14 when she started. Such a long study has never been done before.

As Freeman records, some gifted children live up to their promise and excel, some flunk or drop out or can't keep up. Some escape pushy parents, some travel the world and become famous. Being gifted, she shows, doesn't stop you from getting cancer or becoming HIV positive.

But why is it important to study gifted children? If they are so smart, surely they don't need help? Freeman disagrees forcefully. "Gifted children are always pushed aside as being not worth investigating. But a good deal of our future depends on bright people. If you are gifted, you have the potential to work at a very high level but [there is a danger that] no one will see you – no one will give you what you need."

The number of children classed as gifted is rising and they are being recognised at a younger age.

When she started her work in 1974, Freeman took nobody else's word for it that the children were gifted: she tested them all, independently, herself. She found that they did have "extraordinary abilities", as she puts it. But children can be gifted at things you can't take public exams in, such as empathy or emotional intelligence. It's not all about maths. One woman in Freeman's book – Suzanne – was the person everyone went to with their problems, who was mature beyond her years. She now works with "people in dire straits", says Freeman. "In worldly terms we might not call her a marvellous success." But Suzanne is extraordinarily gifted at what she does.

We rarely hear about gifted people except when something goes wrong – the ones who implode under pressure, become prostitutes in rebellion or never talk to their parents again. In truth, many lead ordinary lives. No matter what they excelled at – music, maths, singing or problem-solving – all these children had extremely high IQ scores. They also worked extremely hard.

What they seem to need is recognition and resources from teachers, love, support and understanding from families. There are tales in Freeman's book of teachers squashing precociousness, of being so incredulous about a pupil's talent that they have ripped to shreds the child's work in front of the whole class. Some people think the gifted need to be taken down a peg or two, not realising that they are doing what comes naturally. While it's considered unacceptable today to heap scorn on a child who is slower than average, a bright one gets no such protection.

Freeman didn't intend to study the children into adulthood, but found that she couldn't stop. "Curiosity made me study them for longer. Every 10 years I'd think it was done and dusted, but because I had access to them and kept in touch with so many of them, from a research point of view it seemed criminal not to carry on."

She is keen to lay to rest another stereotype: that gifted children are tortured, unable to fit in and have emotional problems. In her book, it seems that those who did have problems socially, did so because the parents decided this is how they should be, that their gifted children had to conform to type by having few, or no friends. The children "whose parents had identified them as gifted had emotional problems far more frequently than identically gifted but unlabelled children." So would it be better not to label them? "It's difficult, because you have to do some identification to give them what they need to reach their potential."

Freeman isn't sure why there seem to be more gifted children than before, but hazards a guess. "With IT, children have an active way of learning. You think in different ways."

Does she have advice for a gifted child? "Relax and enjoy life, but hard work is important if you want to realise your potential. To some extent you've got to follow your heart." And for parents? "The love parents give should be without strings, not dependant on achievement. Learn with your child. Don't send them out to name flowers – go out with them. Discover things together. This is particularly important in the early years. It's much more effective to learn with them." Was Freeman herself a gifted child? She seems embarrassed by the question. "I probably was, but we didn't talk about such things in those days."

Gifted Lives: What Happens when Gifted Children Grow Up, by Joan Freeman, published by Routledge, £9.95. To order a copy for £8.95 with free UK p&p, go to guardian.co.uk/bookshop or call 0330 333 6846